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In this paper, we obtain several new intrinsic and extrinsic differentialsphere theorems via Ricci flow.  For intrinsic case, we show that a closed simply connected $n(\ge4)$-dimensional Riemannian  manifold $M$ is diffeomorphic to $S^n$  if one of the following  conditions holds pointwisely: $$  (i)\ R_0>\left(1-\frac{24(\sqrt{10}-3)}{n(n-1)}\right)K_{max};\quad  \ (ii)\\frac{Ric^{[4]}}{4(n-1)}>\left(1-\frac{6(\sqrt{10}-3)}{n-1}\right)K_{max}.$$  Here $K_{max}$, $Ric^{[k]}$ and $R_0$ stand for the maximal sectionalcurvature,  the $k$-th weak Ricci curvature and  the normalized scalar curvature.  For extrinsic case, i.e., when $M$ is a closed simply connected $n(\ge4)$-dimensional submanifold immersed in $\bar{M}$.  We prove that $M$ is diffeomorphic to $S^n$ if it satisfies some pinchingcurvature conditions. The only involved extrinsic quantities in our pinchingconditions are the maximal sectional curvature $\bar K_{max}$ and the squarednorm of mean curvature vector $\vert H\vert^2$. More precisely,  we show that $M$ is diffeomorphic to $S^n$  if one of the following conditions  holds: \begin{itemize}  \item[(1)] $R_0\ge \left(1-\frac{2}{n(n-1)}\right)\bar{K}_{max}+\frac{n(n-2)}{(n-1)^2}\vert H\vert^2$, and strict  inequality is achieved at some point;  \item[(2)] $\dfrac{Ric^{[2]}}{2}\ge (n-2)\bar K_{max}+\frac{n^2}{8}\vertH\vert^2,$ and strict  inequality is achieved at some point;  \item[(3)] $\dfrac{Ric^{[2]}}{2} \ge\frac{n(n-3)}{n-2}\left(\barK_{max}+\vert H\vert^2\right),$ and strict  inequality is achieved at some point.  \end{itemize}  It is worth pointing out that, in the proof of extrinsic case, we applysuitable complex orthonormal frame  and simplify the calculations considerably.  We also emphasize that both of the pinching constants in (2) and (3) areoptimal for $n=4$.

Some sharp differential sphere theorems for nonnegative scalar curvature manifolds